3 . If f ( x ) and g ( x ) are strictly increasing ( or decreasing ) function on [ a , b ], then gof ( x ) is strictly increasing ( or decreasing ) function on [ a , b ].
4 . If one of the two functions f ( x ) and g ( x ) is strictly increasing and other a strictly decreasing , then gof ( x ) is strictly decreasing on [ a , b ].
5 . If f ( x ) is continuous on [ a , b ], such that f ‘ ( c ) ≥ 0 ( f ‗ ( c ) > 0 ) for each c ∈ ( a , b ) is strictly increasing function on [ a , b ].
6 . If f ( x ) is continuous on [ a , b ] such that f ‗( c ) ≤ ( f ‗ ( c ) < 0 ) for each c ∈ ( a , b ), then f ( x ) is strictly decreasing function on [ a , b ].
Maxima and Minima of Functions 1 . A function y = f ( x ) is said to have a local maximum at a point x = a . If f ( x ) ≤ f ( a ) for all x ∈ ( a – h , a + h ), where h is somewhat small but positive quantity .
The point x = a is called a point of maximum of the function f ( x ) and f ( a ) is known as the maximum value or the greatest value or the absolute maximum value of f ( x ). 2 . The function y = f ( x ) is said to have a local minimum at a point x = a , if f ( x ) ≥ f ( a ) for all x ∈
( a – h , a + h ), where h is somewhat small but positive quantity .